Astronomy
&
Astrophysics
A&A 433, 247–265 (2005)
DOI: 10.1051/0004-6361:20042029
c ESO 2005
Models of accreting gas giant protoplanets in protostellar disks
J. C. B. Papaloizou and R. P. Nelson
Astronomy Unit, Queen Mary, University of London, Mile End Rd, London E1 4NS, UK
e-mail: [email protected]
Received 20 September 2004 / Accepted 14 December 2004
Abstract. We present evolutionary models of gas giant planets forming in protoplanetary disks. We first consider protoplanet
models that consist of solid cores surrounded by hydrostatically supported gaseous envelopes that are in contact with the
boundaries of their Hill spheres, and accrete gas from the surrounding disk. We neglect planetesimal accretion, and suppose
that the luminosity arises from gas accretion alone. This generally occurs on a long time scale which may be comparable to the
protostellar disk lifetime. We classify these models as being of type A, and follow their quasi static evolution until the point of
rapid gas accretion is reached.
We consider a second class of protoplanet models that have not hitherto been considered. These models have a free surface,
their energy supply is determined by gravitational contraction, and mass accretion from the protostellar disk that is assumed to
pass through a circumplanetary disk. An evolutionary sequence is obtained by specifying the accretion rate that the protostellar
disk is able to supply. We refer to these models as being of type B. An important result is that these protoplanet models contract
quickly to a radius ∼2 × 1010 cm and are able to accrete gas from the disk at any reasonable rate that may be supplied without
any consequent expansion (e.g. a Jupiter mass in ∼few ×103 years, or more slowly if so constrained by the disk model). We
speculate that the early stages of gas giant planet formation proceed along evolutionary paths described by models of type A,
but at the onset of rapid gas accretion the protoplanet contracts interior to its Hill sphere, making a transition to an evolutionary
path described by models of type B, receiving gas through a circumplanetary disk that forms within its Hill sphere, which is in
turn fed by the surrounding protostellar disk.
We consider planet models with solid core masses of 5 and 15 M⊕ , and consider evolutionary sequences assuming different
amounts of dust opacity in the gaseous envelope. The initial protoplanet mass doubling time scale is very approximately
inversely proportional to the magnitude of this opacity. Protoplanets with 5 M⊕ cores, and standard dust opacity require ∼3 ×
108 years to grow to a Jupiter mass, longer than reasonable disk life-times. A model with 1% of standard dust opacity requires
∼3 × 106 years. Rapid gas accretion in both these cases ensues once the planet mass exceeds 18 M⊕ , with substantial time
spent in that mass range.
Protoplanets with 15 M⊕ cores grow to a Jupiter mass in ∼3 × 106 years if standard dust opacity is assumed, and in ∼105 years if
1% of standard dust opacity is adopted. In these cases, the planet spends substantial time with mass between 30−40 M⊕ before
making the transition to rapid gas accretion. We emphasize that these growth times apply to the gas accretion phase and not to
the prior core formation phase.
According to the usual theory of protoplanet migration, although there is some dependence on disk parameters, migration in
standard model disks is most effective in the mass range where the transition from type A to type B occurs. This is also the
transitional regime between type I and type II migration. If a mechanism prevents the type I migration of low mass protoplanets,
they could then undergo a rapid inward migration at around the transitional mass regime. Such protoplanets would end up in
the inner regions of the disk undergoing type II migration and further accretion potentially becoming sub Jovian close orbiting
planets. Noting that more dusty and higher mass cores spend more time at a larger transitional mass that in general favours
more rapid migration, such planets are more likely to become close orbiters.
We find that the luminosity of the forming protoplanets during the later stages of gas accretion is dominated by the circumplanetary disk and protoplanet-disk boundary layer. For final accretion times for one Jupiter mass in the range 105−6 y, the luminosities
are in the range ∼10−(3−4) L and the characteristic temperatures are in the range 1000−2000 K. However, the luminosity may
reach ∼10−1.5 L for shorter time periods at the faster rates of accretion that could be delivered by the protoplanetary disk.
Key words. accretion, accretion disks – solar system: formation – stars: planetary systems
1. Introduction
Planets are believed to form out of protostellar disks by either gravitational instability (Cameron 1978; Boss 1998) or by
a process of growth through planetesimal accumulation followed, in the giant planet case, by gas accretion (Safronov
1969; Wetherill & Stewart 1989; Mizuno 1980). It is the latter mechanism that we consider in this paper.
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J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
The process is presumed to begin with the accumulation of
the solid cores by the accretion of planetesimals typically exceeding a kilometer in radius which have been formed through
the collisional growth and sedimentation of dust grains in the
protoplanetary disk (see Lissauer 1993, and references therein).
Once the solid core becomes massive enough a significant
gaseous atmosphere forms. The mass required depends to some
extent on physical conditions in the disk, the rate of planetesimal accretion and the dust opacity but is typically several earth
masses (e.g. Mizuno 1980; Stevenson 1982; Bodenheimer &
Pollack 1986). This is consistent with models of Jupiter which
indicate that it has a solid core typically of this magnitude
(Podolak et al. 1993). We note, however, that more recent models suggest that Jupiter’s core may be less massive than previously thought (Saumon & Guillot 2004). Models of Saturn still
indicate a core mass of 10 M⊕ .
During the early build up of the core the luminosity is due
to the liberation of gravitational energy by accreting planetesimals. However, once the mass of the gaseous envelope starts to
become significant the gravitational settling of the gas becomes
important and at some cross over point becomes dominant
(Pollack et al. 1996). At this point models assuming strict thermal equilibrium break down. This is manifest through the fact
that for fixed luminosity due to planetesimal accretion, there
is a maximum or critical core mass for which a strict thermal
equilibrium model can be constructed (see e.g. Papaloizou &
Terquem 1999). Beyond this point the evolution is no longer in
thermal equilibrium and if the protoplanet remains in contact
with adequate disk material, gas accretion may ensue.
The purpose of this paper is to examine the protoplanet
models subsequent to the attainment of the critical core mass
in the context of the protoplanetary disk environment and disk
planet interactions. We assume that the core becomes isolated
from further planetesimal accretion so that settling of accreted
gas is the only energy source. The rationale for this assumption is discussed in Sect. 4.2. We consider two types of model.
The first type, which we denote as type A, is fully embedded
in the protostellar disk and hence has an effective radius equal
to that of the Roche lobe or Hill sphere. This is the correct radius to use rather than the Bondi radius which is never significantly smaller for any of the models we study. At some mass,
these models tend to enter a rapid accretion phase. This occurs
when the planet mass ∼0.1 MJ , MJ denoting a Jupiter mass,
and is similar to that for which either significant perturbation
to the protoplanetary disk through local mass accretion or diskplanet interaction begins (e.g. Nelson et al. 2000). These processes eventually lead to gap formation. Accordingly we consider models of a second type, type B, which are no longer
enveloped at the Roche lobe but are assumed to have a free
surface and accrete from a circumplanetary disk at a rate determined by the external throughput from the protostellar disk. We
find that these can be constructed for a wide range of accretion
rates indicating that during the later stages a forming protoplanet can comfortably accrete at any rate reasonably supplied
by the protostellar disk.
We supplement these models of protoplanet evolution with
hydrodynamical simulations of the interaction between low
mass protoplanets and protostellar disks. The purpose of these
models is to establish plausible accretion times scales for the
freely accreting protoplanet models of type B.
This paper is organised as follows. We present the basic
equations for the protoplanet models in Sect. 2, and discuss
the appropriate boundary conditions in Sect. 3. In Sect. 4 we
describe how evolutionary sequences are constructed for protoplanet models of type A and B, accounting for gas accretion
from the protostellar disk. We discuss the numerical procedure
adopted for the hydrodynamic simulations of disk-planet interactions in Sect. 5. The results of our calculations are presented in Sect. 6, and their implications are discussed in Sect. 7.
Finally we draw our conclusions in Sect. 8.
2. Basic equations for the protoplanet models
We adopt the approach of previous workers (e.g.
Bodenheimer & Pollack 1986; Pollack et al. 1996) and
approximate the protoplanetary structure as being spherically
symmetric, the approach being similar to that followed in
modeling stellar structure. Many of the details are given in
Papaloizou & Terquem (1999).
The interior state variables at any point in a model are functions only of the distance to the centre, r, also characterized as
the spherical polar radius. We assume the models are in hydrostatic equilibrium and neglect rotation. The equation of hydrostatic equilibrium is
dP
= −ρg.
dr
(1)
Here, the pressure is P, the local acceleration due to gravity
is g = GM(r)/r2 with M(r) being the mass, including that of
any solid core, interior to radius r and G is the gravitational
constant. The mass interior to radius r satisfies
dM
= 4πr2 ρ,
dr
(2)
where ρ is the density.
For the calculations presented here, we adopt the equation
of state for a hydrogen and helium mixture given by Chabrier
et al. (1992). The mass fractions of hydrogen and helium are
taken to be 0.7 and 0.28, respectively. The luminosity Lrad
transported by radiation satisfies
Lrad
4acT 3 dT
,
=−
2
3κρ dr
4πr
(3)
where a, c, T and κ are the radiation constant, the speed of
light, the temperature and the opacity respectively. The calculations reported here are based on the opacities given by Bell &
Lin (1994). These, being functions of density and temperature,
include contributions from molecules, atoms, ions and dust
grains. The latter produce an increase in the opacity amounting to several orders of magnitude for T < 1600 K.
2.1. Inner convective regions
Most of the gas mass within the models is unstable to convection so a theory of energy transport by convection is needed.
J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
249
We adopt the conventional mixing length theory (e.g. Cox &
Giuli 1968).
The radiative and adiabatic temperature gradients ∇rad and
∇ad are defined through
∂ ln T
3κLr P
=
,
(4)
∇rad =
∂ ln P rad 16πacGMT 4
3.2. The outer boundary
and
For these models, subsequently denoted as of type A, we assume the structure extends to the Roche lobe or boundary of
the Hill sphere beyond which material must be gravitationally
unbound from the protoplanet. For this radius we adopt
1/3
2 Mpl
rL =
Rp ,
(8)
3 3M∗
∇ad =
∂ ln T
∂ ln P
,
(5)
S
with the subscript S denoting evaluation at constant entropy.
∇ad is a quantity determined directly from the thermodynamics
of the equation of state alone.
The total luminosity is Lr . During the phase of solid core
growth it is expected that this is produced by the gravitational energy of accreting planetesimals (e.g., Mizuno 1980;
Bodenheimer & Pollack 1986). However, for the later phases
considered here, the source of energy is primarily settling and
accretion of gas (see Sect. 4.2).
When ∇rad < ∇ad , the gas is convectively stable and the
energy is transported entirely by radiation. On the other hand
when ∇rad > ∇ad , the medium is convectively unstable and
some of the energy is transported by convection. We write
the total luminosity passing through a sphere of radius r as
Lr (r) = Lrad + Lconv , where Lconv is the luminosity associated
with convection. Adopting the mixing length theory (Cox &
Giuli 1968) we have
3/2
∂T
∂T
−
Lconv = πr2C p Λ2
∂r S
∂r
1 ∂ρ ρg ×
(6)
,
2 ∂T P where Λ = |αP/(dP/dr)| is the mixing length, α being a constant parameter expected to be of order unity, (∂T/∂r)S =
∇ad T (d ln P/dr), and the subscript P denotes evaluation at constant pressure. We adopt the mixing length parameter α = 1.
3. Boundary conditions
3.1. The inner boundary
We assume that there is a solid core of mass Mcore with a
uniform mass density ρcore = 3.2 g cm−3 (e.g. Papaloizou &
Terquem 1999). The boundary condition the models that calculate the structure of the gaseous envelope must satisfy is that
the total mass M(rcore ) = Mcore when
1/3
3Mcore
·
(7)
r = rcore =
4πρcore
For models with no source of accretion energy at the core surface, such as those considered here, we also require Lr = 0.
However, the model interiors are convective with radiation
making negligible contribution to the heat transport, accordingly adiabatic stratification is a good approximation near the
inner boundary independently of any reasonable value for Lr ,
with the consequence that we do not actually need to enforce
the condition Lr = 0 there.
We here consider two different classes of model which have differing boundary conditions. We consider each of these in turn.
3.2.1. Enveloped models in contact with the roche lobe
where Mpl is the total planet mass including gas and solid
core and Rp is the orbital radius or distance of the protoplanet,
assumed in circular orbit, from the central star. The structure state variables are assumed to eventually join smoothly to
those associated with the enveloping protoplanetary disk where
T = T d , P = Pd and ρ = ρd , respectively.
Thus the boundary conditions are that at r = rL , M(rL ) =
Mpl , P = Pd and the temperature is given by
1/4
4
T = T d4 + T effp
,
(9)
4
where T effp
= 3τL L/(4πacrL2 ), with L denoting the total luminosity escaping from the surface.
Here we approximate the additional optical depth above
the protoplanet atmosphere, through which radiation passes, by
(Papaloizou & Terquem 1999)
τL = κ (ρd , T d ) ρd rL .
(10)
This expresses the fact that T must exceed T d at r = rL in order
that the luminosity be radiated away from the protoplanet into
the surrounding disk. In practice it is found for the models here
that T always only slightly exceeds T d – i.e. T effp is effectively
small at r = rL (see Figs. 3 and 4 below).
For most models we adopt disk parameters appropriate to
5 AU from the disk model of Papaloizou & Terquem (1999)
with Shakura & Sunyaev (1973) α = 0.001 and steady state
accretion rate of 10−7 M⊕ /yr. Accordingly T d = 140.047 K
and Pd = 0.131 dyn cm−2 .
3.2.2. Models with free boundary accreting
from the protostellar disk
In contrast to embedded models of type A, we can consider
models that have boundaries detatched from and interior to the
Roche lobe which still accrete material from the external protoplanetary disk that orbits the central star. This is expected as numerical simulations of disk planet interactions have shown that
once it becomes massive enough a protoplanet forms a gap in
the disk but is still able to accrete from it through a circumplanetary disk (see e.g. Kley 1999; Nelson et al. 2000; Lubow et al.
1999). We thus consider models with free boundaries which
are able to increase their mass and liberate gravitational energy
through its settling. We subsequently refer to such models as
type B.
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J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
For these models, in contrast to those of type A, the effect
of the exterior disk material on the surface boundaries is small.
Thus for the boundary condition on T we again adopt Eq. (9)
but with τL = 0.5. We note for these models T effp is in general
significantly larger than T d . For the boundary condition on P
we adopt
do form a one parameter family specified by Mpl . Accordingly
we may write E = E(Mpl ).
When such a model increases its mass slightly so that
Mpl → Mpl + dMpl , the change of energy content is dE =
(dE/dMpl)dMpl . If the energy change balances losses by radiation in time dt, Ldt, then conservation of energy requires that
g
P = Pd + ,
κ
dE dMpl
= −L,
dMpl dt
(11)
which is the conventional stellar structure boundary condition
(e.g. Schwarzschild 1958) but with the addition of the background pressure Pd , which in fact for these models makes only
a small contribution.
In order to have a complete system for which the evolution can be calculated Eqs. (1)−(6) need to be supplemented
by an equation governing internal energy production and the
internal luminosity, normally the first law of thermodynamics.
Here we simplify matters by using the fact that most of the internal energy of the models is contained within a deep convection zone. The thermal time scale associated with relaxation of
the exterior layers is expected to be much shorter than the thermal relaxation time scale associated with the model as a whole.
Under these conditions, if the model evolution time scale is on
the global Kelvin-Helmholtz time scale or longer, it is a reasonable assumption that Lr is constant in the outer layers. Because
of efficient convection the inner convection zone is unaffected
by the distribution of Lr . Accordingly we make the assumption
that Lr = L is constant in the outer layers. This is expected
to hold during the longest lasting evolutionary phases for all
masses and and at all times for the larger masses which tend
to have only a very thin surface radiative shell, but we bear in
mind that it may fail when the evolution time becomes very
short. With the above assumption we obtain a complete system
for which the evolution may be calculated.
4. Accretion, settling and evolutionary sequences
The models we consider here provide an evolutionary sequence
with their mass Mpl increasing through accretion from the protoplanetary disk. This material also liberates gravitational energy as it settles. To describe their evolution we consider the
total energy of the protoplanet within the Roche lobe
Mpl GM dM.
(12)
U−
E=
r
0
Here U is the internal energy per unit mass and we neglect the
energy involved in bringing material from ∞ to the Roche lobe.
This is justified because most of the mass is concentrated well
inside it where the specific energies are much higher.
4.1. Models of type A
We now consider models of type A. From the theory of stellar
structure, if the source of energy was specified, a model would
be uniquely determined once Mpl is specified. Not specifying
the source of energy leaves one free parameter. However, because the radius is specified as a function of Mpl through the
Roche lobe condition, this freedom is lost so that the models
(13)
with L being the luminosity at the surface. This determines the
evolution of models of type A.
4.2. Models of type B
In contrast to models of type A, for an assumed externally supplied accretion rate, models of type B form a two parameter
family in that, without specification of the energy source, and
given their freedom to determine their own radius, they require
specification of both Mpl and L in order for a model to be constructed. Thus E = E(Mpl , L). Accordingly for small changes
in mass, and luminosity the change in E is
∂E
∂E
dE =
dMpl +
dL.
(14)
∂Mpl
∂L
Now for these models, matter is presumed to join the protoplanet on its equator after having accreted through a circumplanetary disk. In this case we assume the accretion rate to be
prescribed by the dynamics of the disk-planet interaction while
gap formation is taking place. This is found to be the case from
simulations of disk-planet interactions where it is found that
an amount of material comparable to that flowing through the
disk may be supplied to the protoplanet (Kley 1999; Nelson
et al. 2000; Lubow Siebert & Artymowicz 1999; and simulations presented in Sect. 6.2).
In arriving there all available gravitational binding energy
of −GMpl /rs per unit mass, rs being the surface radius, has been
liberated and so an amount of energy −GMpl dMpl /rs must be
subtracted from dE in order to obtain the energy available to
replace radiation losses.
Therefore if the changes occur over an interval dt, we must
have dE + GMpl dMpl /rs = (∂E/∂Mpl)dMpl + (∂E/∂L)dL +
GMpl dMpl /rs = −Ldt.
Thus total energy conservation for models of type B enables the calculation of evolutionary tracks through.
GMpl dMpl ∂E dL
∂E
+
= −L.
(15)
+
∂Mpl
rs
dt
∂L dt
Note that as we regard the accretion rate dMpl /dt as specified
for these models, Eq. (15) enables the evolution of L to be calculated.
Thus Eqs. (13) and (15) constitute the basic equations governing the evolution of models of type A and type B respectively.
Note that we neglect any input from planetesimal accretion
during and after the phase when the core becomes critical. The
primary reason for doing this is that we are interested in examining the fastest time scales possible for giant planet formation
J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
via the core instability scenario. The inclusion of planetesimal
accretion and the associated accretion luminosity will have the
effect of lengthening the time scale of formation, provided that
the core mass itself does not increase significantly. However,
there are also uncertainties about how large the planetesimal
accretion rate ought to be.
Previous work on the formation of gas giant planets via the
core instability model assumed that core formation can proceed
through runaway growth in which a protoplanetary core can
grow by accreting essentially all planetesimals in its feeding
zone (e.g. Pollack et al. 1996). This resulted in a core formation
time of ∼few ×105 years. Simulations by Ida & Makino (1993)
indicate, however, that runaway growth slows down prior to the
completion of core formation, and proceeds through a more
orderly mode of planetesimal accretion known as oligarchic
growth. This arises because neighbouring planetary embryos
stir up the random motions of the planetesimal swarm, reducing the effectiveness of gravitational focusing. N-body simulations of protoplanetary core formation indicate that obtaining
cores of the necessary mass is not an easy task to achieve during the oligarchic growth phase, in part due to planetary cores
of a few M⊕ repelling the surrounding planetesimals and opening gaps in the planetesimal disk, and in part due to the excitation of planetesimal eccentricities and inclinations by the
“oligarchs” (e.g. Thommes et al. 2003).
After core formation, and during the longest phase of evolution involving gas settling onto the core, the calculations of
Pollack et al. (1996) result in planetesimal accretion rates that
are a factor of ∼3 times smaller than those of the gas accretion
rate, and this planetesimal accretion results in significant accretion luminosity. This planetesimal accretion arises because
the feeding zone expands as the planet mass increases due to
gas accretion, and depends on the strict assumption that planetesimals are not allowed to enter or leave the feeding zone.
Thus the possibility of gap formation in the planetesimal disk,
as found by Thommes et al. (2003), is not accounted for in
these models, although one may reasonably expect its effect to
be increasingly important as the planet mass increases.
The generation of significant luminosity from planetesimal
accretion depends on where it is assumed the energy is deposited within the protoplanet. Large (100 km) planetesimals
are able to penetrate deep into the planetary interior, and so provide a significant source of energy by virtue of descending deep
into the gravitational potential well. Smaller planetesimals or
fragments are more likely to dissolve higher up in the planet
atmosphere, and so will contribute less accretion luminosity.
A possible resolution of the long time scales of formation for
planetary cores reported by Thommes et al. (2003) is that collisions between planetesimals result in fragmentation when their
random motions are excited by the forming planetary embryos
(e.g. Rafikov 2004). This possible generation of smaller planetesimals results in increased efficiency of gas drag by the nebula in damping random motions, thus speeding up planetesimal
accretion by planetary embryos. This potential modification of
the size distribution will also have an impact on the accretion
luminosity generated by accreted planetesimals.
In the light of these uncertainties in the radial distribution and size distribution of planetesimals, and its effect on
251
planetesimal accretion rates during the gas settling, and rapid
gas accretion phase of giant planet formation, we believe it is
justified to treat the planetesimal accretion rate and its associated luminosity generation as a free parameter of the problem. A similar approach has been argued for by Ikoma et al.
(2000). As we are interested in examining the shortest possible
time scales for giant planet formation, we neglect the effects of
planetesimal accretion in this study.
With the above assumption we have a complete system of
Eqs. (1)−(6), (13) and (15) for which the evolution may be
calculated.
5. Disk-planet simulations
In order to estimate the rate at which an accreting protoplanet
can accrete gas from a protoplanetary disk, we performed
hydrodynamic simulations of low mass protoplanetary cores
embedded in viscous disk models. These simulations were
performed with a modified version of the grid based hydrodynamics code NIRVANA (Ziegler & Yorke 1996).
5.1. Initial setup and boundary conditions
The disk models are simple 2-D models with the initial surface
density given by a power law Σ(R) = Σ0 R−1 . We set Σ0 such
that there are 0.02 M interior to 40 AU, similar to the minimum mass solar nebula model. We assume a locally isothermal equation of state, and specify that the disk vertical thickness to radius ratio have a constant value H/R = 0.05. We
model the angular momentum transport processes in the disk
using a simple “alpha” prescription for the disk viscosity in the
Navier-Stokes equation – i.e. the kinematic viscosity is given
by ν = αcs H where α is a parameter, cs is the sound speed, and
H is the local disk thickness. We consider values of α = 5×10−3
and 10−3 .
The number of grid cells used was (NR , Nφ ) = (260, 630).
The inner boundary of the computational domain was placed
at R = 0.4 and the outer boundary at R = 3. Reflecting boundary conditions were used at both radial boundaries, and linear
viscosity was used between 0.4 ≤ R ≤ 0.6 and 2.5 ≤ R ≤ 3
to reduce reflection of waves excited by the planet. The gravitational potential of the planet was softened using a softening parameter b = 0.5H(R p) – i.e. half of the local disk
semi-thickness.
Simulations were initiated by placing a low mass planet (either 15 or 30 Earth masses) at a radius R p = 1 in the disk. The
planet was assumed to accrete gas that entered its Hill sphere.
This was achieved by removing gas from any cells that lay
within half of the planet Hill sphere. The e-folding
time of this
gas removal was τacc = Ω−1 , where Ω = (GM∗ )/R3p . Thus
this corresponds to the extreme case when the planet accretes
material within the Hill sphere on the dynamical time scale.
The gas that was removed from the Hill sphere was added to the
planet at each time step, such that the planet mass is a function
of time. Similar models are described in Nelson et al. (2000).
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J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
6. Protoplanet model calculations
We solve Eqs. (1)−(6) with the boundary conditions described
above to get the structure of the protoplanet models.
For a fixed accretion rate onto a core Ṁcore at a given radius,
there is a critical core mass Mcrit above which no solution can
be found in hydrostatic and thermal equilibrium that joins on
to the protoplanetary disk model assumed at the Roche lobe.
In this paper we consider cores with Mcore = 15 M⊕ and
Mcore = 5 M⊕ . Our evolutionary calculations commence close
to the state when the cores are critical, that is no further gas
can be added in strict hydrostatic equilibrium. At this stage the
evolution is slowest and we compute a type A model sequence
by use of Eq. (13). These models are in contact with the Roche
lobe.
We also construct type B model sequences. These satisfy
the free surface boundary conditions given by Eqs. (9) and (11).
Because the surface is free, these form a two parameter sequence in that evolutionary tracks for a given mass can be
started for a range of radii (or equivalently the luminosity may
be used as a parameter). This the same situation as in standard
pre-main sequence contraction where a stellar model of a given
mass can be started at different points on an evolutionary track
corresponding to different radii.
We have considered models using the Bell & Lin (1994)
opacities hereafter referred to as standard. These have a very
large contribution from dust grains for T < 1600 K and because there is clearly some uncertainty about the disposition
of the dust particularly under circumstances where the protoplanet is assumed isolated from further planetesimal accretion,
we have explored the effect of reducing this contribution to the
opacity by factors of up to 100 for both models of type A and B.
We have done this, by making the reduction for the opacity as
a whole, for T < 1600 K only, and with a reduction factor that
is constant for T < 1600 K and which then decreases linearly
to unity at T = 1700 K. In practice we find that the results are
essentially independent of whether such a linear join is made
or not. The uncertainty in the magnitude of the surface opacity as well as its important role in controlling the evolutionary
time scale of an embedded protoplanet has been pointed out by
Ikoma Nakazawa & Emori (2000).
6.1. Models of type A
We begin by describing some typical models of type A. In
Fig. 1, state variables are plotted for a protoplanet model with
Mcore = 15 M⊕ which has a total mass 25.3 M⊕ . As expected
the deep interior of this model is convective with little energy
transported by radiation. However, there are two convective regions which occur for 679 K > T > 263 K and T > 2100 K.
Approximately ninety eight percent of the mass is in the inner
convective zone. This means that most of the thermal inertia
is contained within the deep convection zone rendering the assumption of little spatial variation of the luminosity in the upper layers a reasonable approximation. The existence of two
separate convective regions is in contrast to what we find for
models of type B that approach 1 MJ . In those cases we find
a single interior convection zone for T > 1000−2000 K, with
negligible mass in the outer radiative region.
In Fig. 2, we illustrate the behaviour of the state variables
for a protoplanet model with the smaller 5 M⊕ core mass. The
total mass is 17.6 M⊕ . This has similar properties to the previous case regardless of the fact that the core mass is three times
smaller. Convective heat transport occurs when 720 K > T >
264 K and when T > 2100 K. The inner eighty percent of the
mass is convective.
In Fig. 3 we illustrate the evolution of models of type A for
Mcore = 5 M⊕ . Cases with standard opacities and with opacity
reductions of three, ten and one hundred made globally and
for T < 1600 K are shown. In all cases as the models gain in
mass from the protoplanetary disk their luminosity increases
and their evolutionary time measured through their accretion
time Mpl /(dMpl /dt) decreases. In the standard opacity case the
accretion time is very long, exceeding 108 y. However, this time
reduces by the opacity reduction factor independently of where
this is applied even though the situation might have appeared
to have been complicated by the existence of two convective
regions. Thus times ∼3×106 y are attained for reduction factors
of one hundred.
The evolutionary time scale of the models begins to decrease rapidly once Mpl ∼ 20 M⊕ , becoming less that 105 y
even for the standard opacity case. This phenomenon, which is
sensitive to relatively minor model details, can be traced to the
fact that dE/dM becomes small or that less and less binding
energy is liberated as the mass increases. This is likely to indicate the onset of a rapid collapse and possible detachment from
the Roche lobe. As the effect of disk planet interactions and local gas depletion are likely to become important, we have not
tried to follow such rapid evolution with the simplistic models adopted here. Rather we have considered evolutionary sequences of type B which are likely to be the outcome. Although
the position where such a sequence is joined cannot of course
be determined without considering the above rapid phase of
evolution.
Figure 3 also shows that T effp , the effective temperature
needed to supply the luminosity of the model, is always small
when compared to the surrounding protostellar disk temperature. This indicates negligible thermal perturbation of the protostellar disk.
In Fig. 4 we illustrate the evolution of models of type A for
but for two models with standard opacity Mcore = 15 M⊕ . For
these models the longest evolutionary times are in the 3 × 106 y
range. The attainment of short evolutionary times likely leading to detachment from the Roche lobe occurs for Mpl ∼ 35 M⊕
in this case. The two models illustrated differ in surface boundary conditions. The model illustrated with dotted curves is embedded in a disk with the same temperature but with a density
ten times larger than usual. Except during the beginning of the
rapid evolution phase the models show very similar behaviour.
In Fig. 4 we also plot evolutionary tracks for which the
opacity was reduced by factors of ten and one hundred in the
surface layers for which T > 1600 K with a linear transition
to standard opacities occurring for 1700 K > T > 1600 K.
For these sequences the transition mass is unaffected but the
evolutionary time scales are factors of three and thirty faster
J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
253
Fig. 1. State variables are plotted for a protoplanet model of total mass 25.3 M⊕ with a 15 M⊕ solid core. Apart from temperature in K, these
are given in cgs units. The upper left panel gives a plot of interior radius as a function of interior mass. The upper right panel gives a pressure
temperature plot. The lower left panel gives a plot of temperature as a function of interior mass. The lower right panel gives a plot of the local
density as a function of interior mass. Convective heat transport occurs when 679 K > T > 263 K and when T > 2100 K. Approximately the
gas component of the inner ninety eight percent of the mass is convective.
respectively. This means that the formation time scale is reduced to ∼105 y in the latter case.
6.2. Planet accretion rates
As described in Sect. 4.2, the planetary models that we construct with a free surface require a gas accretion rate to be specified. We have performed simulations of low mass protoplanets
embedded in protoplanetary disks to estimate the accretion rate
onto a protoplanet that may be supplied by a protoplanetary
disk, using different assumptions about the initial planet mass
and disk viscosity. The results of these simulations are presented in Fig. 5. The left hand panel shows the evolution of the
planet mass, and the right hand panel shows the accretion rate
as a function of time. We considered initial protoplanet masses
of Mpl = 15 and 30 Earth masses, and viscosities with α = 10−3
and α = 5 × 10−3. The solid line in Fig. 5 shows the model with
Mpl = 15 M⊕ and α = 10−3 . The dashed line shows the model
with Mpl = 30 M⊕ and α = 10−3 . The dotted line shows the
model with Mpl = 15 M⊕ and α = 5 × 10−3, and the dot-dashed
line shows the model with Mpl = 30 M⊕ and α = 5 × 10−3 .
It is clear that the initial mass assumed for the protoplanet is
unimportant as the models quickly converge. However, quite
differing evolutionary sequences are obtained as a function of
disk viscosity. For higher viscosity, a protoplanet that is thermodynamically permitted to accrete gas rapidly from a disk
can grow to become a Jupiter mass in around 3000 years. For
lower viscosity the growth time can be extended considerably,
with an α = 10−3 requiring a time ≥2 × 104 years for a Jupiter
mass planet to form. The simulations presented here are too
low in resolution to model the circumplanetary disk that is expected to form around the accreting protoplanet or its interaction with it, and in principle the requirement that material
accrete through this circumplanetary disk before reaching the
planet surface could act as a bottle neck and significantly extend these accretion time scales. However, simple estimates of
the accretion time through such a circumplanetary disk, and the
accretion rates obtained from high resolution 3-D simulations
(e.g. D’Angelo et al. 2003) suggest that this is not the case.
These high resolution simulations indicate that the circumplanetary disk that forms within the planetary Hill sphere has a radius 2RH /3 where RH = Rp (Mpl /3 M∗ )1/3 is the Hill sphere
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J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
Fig. 2. As in Fig. 1, but for a protoplanet model of total mass 17.6 M⊕ with a 5 M⊕ solid core. Convective heat transport occurs when 720 K >
T > 264 K and when T > 2100 K. Approximately the gas component of the inner eighty percent of the mass is convective.
radius. The viscous evolution time through a disk of such radius is
3 2
2 RH
τν 3
ν
where ν = αcs H is the kinematic viscosity, cs being the sound
speed at the outer edge of the circumplanetary disk, and H being the disk semi-thickness there. The viscous time scale may
be expressed in units of the planet orbital period as
3
(3.2)−1 2 RH 2
·
τν 2πα 3
H
We note that RH /H ≡ M, where M is the Mach number
of the flow in the outer regions of the circumplanetary disk.
Simulations by D’Angelo et al. (2003), that account for heating
and cooling of the circumplanetary disks, results in Mach numbers of M < 2 in their outer parts, indicating that these disks
are rather thick. If we adopt the values of α used in the numerical simulations, we estimate ‘that the accretion time through
the circumplanetary disk is τν < 141 yr for α = 5 × 10−3 , and
τν < 707 yr for α = 1 × 10−3 . These time scales should be
compared with the mass accretion times presented in Fig. 5.
For the α = 5 × 10−3 runs the mass doubling time scale is
found to be ≈500 yr. For α = 1 × 10−3 this accretion time is
≈1000 yr. This suggests that the simple prescription for modeling mass accretion in the simulations does not significantly
affect the long term accretion times presented, as the protostellar disk supplies mass to the protoplanet on time scales longer
than reasonable accretion times through the circumplanetary
disk. We note, however, that more detailed 3D simulations including a proper account of the thermodynamic evolution of
the gas will be required to definitively settle this question.
The accretion times obtained in Fig. 5 range from a few
thousand years to a few tens of thousands of years, and show
that the actual accretion time obtained is sensitive to the disk
viscosity assumed. In the planet models of type B presented
below, we consider accretion times of between 5 × 103 to
9 × 105 years. These cover the accretion times obtained for a
protoplanet on short time scales in the extreme case when it
is immersed ab initio into an unperturbed disk as in the above
simulations. They also allow for the situation where there is gas
depletion such that the protoplanet can only accrete for longer
J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
255
Fig. 3. This figure illustrates the evolution of protoplanet models which maintain contact with the protoplanetary disk and fill their Roche lobes
while they accrete from it. They have fixed solid core masses of 5 M⊕ and are situated at 5 AU. The upper left panel shows Luminosity in cgs
units as a function of their increasing mass, M, in earth masses. The upper right panel shows the gas accretion rate in M⊕ y−1 as a function of
mass while the lower left panel gives the accretion time M/ Ṁ in yr as a function of mass. The lower right panel gives the temperature T effp (see
text) as a function of mass. The models shown have standard opacities (full line), standard opacities reduced by a factor of three (dotted line),
standard opacities reduced by a factor of ten (dashed line), and standard opacities reduced by a factor of one hundred (triple dot dashed line)
respectively. In addition models with an opacity reduction of a factor of one hundred applied only for T < 1600 K are plotted (upper full line
in all panels except lower left where it is the lower full line).
time scales at an assumed mass flow rate through the protostellar disk of 10−9 M y−1 .
6.3. Models of type B
Figure 6 illustrates the evolution of protoplanet models which
accrete from the protoplanetary disk at an assumed fixed rate
of 1 MJ in 9 × 104 y. This accretion rate corresponds to 1.1 ×
10−8 M y−1 which is typical of T Tauri disks, but it leads to
a rapid final accretion time for a Jupiter mass if most of this
is accreted by the protoplanet as is indicated by the simulations of disk-planet interaction presented in Sect. 6.2 (see also
Kley 1999). The models have Mcore = 15 M⊕ . For these models the duration of the evolution is determined by the accretion
rate and is terminated when the protoplanet reaches 1 MJ . The
four models shown correspond to different starting masses and
luminosities. For a given mass is it is possible to start with a
range of luminosities or for the same luminosity it is possible to start with a range of masses. Here the dashed and dotdashed curves correspond to models which start with almost
the same mass but very different luminosities while the dotted
and dashed curves start with the same luminosity but differ in
mass by a factor of 2.5. The resulting evolutionary tracks tend
to show convergence as time progresses especially in the case
of the radius which ends up in the range 2.25 ± 0.75 × 1010 cm
after ∼2 × 104 y. The models attain T eff > 400 K for most of the
evolution. However the luminosity expected from circumplanetary disk accretion 3.26 × 1030 erg s−1 is only approached by
the most luminous model.
Figure 7 illustrates evolutionary tracks for a fixed assumed
accretion rate that is ten times slower. As a consequence of this
the evolution times are ten times longer. The convergence of
the evolutionary tracks is greater in this case with all radii being close to ∼3 × 1010 cm after ∼105 y. The indication is that
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J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
Fig. 4. As in Fig. 3 but for two models with standard opacity and solid cores of 15 M⊕ . The model illustrated with the full curve is embedded
in a standard disk while the model illustrated with a dotted curve is embedded in a disk with the same temperature but with a density ten times
larger. These two protoplanet models show very similar behaviour indicating lack of sensitivity to the detailed boundary conditions. In addition
we illustrate two models with this core mass embedded in a standard disk but with opacities which have a reduction factors of ten and one
hundred (dashed curves and triple dot dashed curves respectively) that is constant for T < 1600 K and which then decreases linearly to unity at
T = 1700 K.
values of T eff ∼ 700−800 K for these models are sustained for
∼106 y. However, the luminosity expected from circumplanetary disk accretion at the later phases ∼3.26 × 1029 erg s−1 (calculated adopting a radius of 2 × 1010 cm for the protoplanet)
is only exceeded at early times by the most luminous model
which then becomes fainter at later times. But note that for this
sequence of models and others presented later, models accreting from the disk can exist which have small luminosities the
circumplanetary disk luminosity and also that due to the protoplanet disk boundary layer, equal to 0.5GMpl /rs (dMpl /dt) (see
e.g. Lynden-Bell & Pringle 1974).
In Fig. 8 we show tracks for an accretion rate of 1 MJ in
9 × 104 y for models with Mcore = 5 M⊕ . The behaviour is similar to that in the higher core mass case. In Fig. 9 we show models with Mcore = 5 M⊕ with the same accretion rate which have
opacities globally reduced by a factor of three and in Fig. 10
the reduction is by a factor of ten. In all of these cases there
is a tendency of the tracks to converge especially the radii of
different models to a value of about 2 × 1010 cm, with the lower
opacity models being slightly smaller. In all cases the protoplanet luminosities is exceeded at late stages by the circumplanetary disk luminosity.
In Fig. 11 we illustrate models with Mcore = 5 M⊕ and standard opacities accreting from the disk at a rate that is ten times
slower while in Fig. 12 the opacity is globally reduced by a factor of three at that accretion rate. In these cases the evolution is
prolonged by a factor of 10. These models are again similar to
the previous ones and indicate that a model starting from one
Saturn mass and radius ∼6 × 1010 cm could sustain effective
temperatures ∼700 K for times approaching 106 y.
Finally in Figs. 13 and 14 we explore models with Mcore =
5 M⊕ subjected to a very high accretion rate from the disk at a
rate of 1 MJ in 5 × 103 y with an opacity reduction by a factor
of ten applied globally in the former case and applied only for
T < 1600 K in the latter. Paradoxically (see the discussion in
Sect. 7) these models may appear somewhat cooler and less
intrinsically luminous than those calculated for lower accretion
J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
257
Fig. 5. This figure shows the mass accretion onto planetary cores obtained from the hydrodynamic simulations described in the text. The left
panel shows accumulated mass onto each planet, and the right hand panel shows the mass accretion rate in units of Jupiter masses per year. We
note that the values obtained for this quantity span the range of values used for the accretion rates onto the detached planet models described in
Sect. 6.3.
Fig. 6. This figure illustrates the evolution of protoplanet models which accrete from the protoplanetary disk at an assumed rate of one Jupiter
mass in 9 × 104 y but are detached from their Roche lobes. They have fixed solid core masses of 15 M⊕ and are situated at 5 AU. The upper
left panel shows Mass in cgs units as a function of time in yr. The upper central panel shows the protoplanet radius in cgs units as a function of
time. The upper right panel shows the protoplanet radius as a function of the total current protoplanet mass. The lower left panel gives the total
intrinsic luminosity of the protoplanet together with a contribution 0.5GMpl /rs (dMpl /dt) which could be due to, either the inner regions of the
circumplanetary disk or the disk protoplanet boundary layer assuming small protoplanet rotation, in cgs units as a function of time. The lower
central panel gives the total intrinsic luminosity of the protoplanet assuming no contribution from the circumplanetary disk or disk-protoplanet
boundary layer. The lower right panel gives the effective temperature as a function of time. The four models shown correspond to differing
initial conditions corresponding to different starting masses and luminosities. The same line type in different panels corresponds to the same
model. The resulting evolutionary tracks tend to show convergence as time progresses.
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J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
Fig. 7. As in Fig. 6 but for an assumed accretion rate from the disk that is ten times slower. As a consequence of that it takes about ten times
longer to attain one Jupiter mass in these cases.
Fig. 8. As in Fig. 6 but for models with a 5 M⊕ solid core.
J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
259
Fig. 9. As in Fig. 6 but for models with a 5 M⊕ solid core which have opacities reduced by a factor of three.
rates. However, convergence of model radii towards 2×1010 cm
again occurs.
7. Discussion
7.1. Planetary growth and migration
In this section we discuss the evolutionary sequences associated with planetary models of type A and B in the context of
disk-planet interactions and planetary migration.
7.1.1. 5 M ⊕ core models
In Fig. 15 we have plotted a schematic diagram showing the
variation of planetary growth times as a function of planet mass
for models with cores of 5 M⊕ . Here we emphasize that the
growth time referred to here and below apply to the gas accretion phase and not to the time required for the solid core
to form. Also plotted in this diagram is a shaded region which
shows the range of T Tauri disk life-times as inferred from infrared observations (e.g. Beckwith et al. 1990; Sicilia-Aguilar
et al. 2004) ranging between 3 × 106 to 3 × 107 years. We have
also plotted migration times appropriate to a standard quiescent disk as a function of planet mass, where the migration
rates are taken from Tanaka et al. (2002). We assume that the
disk surface density scales as Σ(R) ∝ R−1 , that the disk surface density at 5 AU (assumed to be the planet semi major
axis) is Σ(Rp ) = 160 g cm−2 , and that H/R = 0.05 (in other
words the model is similar to the minimum mass solar nebula model). In plotting this diagram we also take account of
the fact that more massive planets begin to open gaps, and
make a transition to type II migration (Ward 1997), for which
Jovian mass planets migrate on the viscous time scale (here
assumed to be 105 years). Also included is a shaded area for
masses between 30 and 100 M⊕ that takes account the possibility of fast or runaway migration for this mass range (Masset &
Papaloizou 2003). This can only occur for disk masses greater
than the minimum mass solar nebula model. We make a rough
estimate of the migration rate associated with runaway migration as being the type I rate for a disk surface density 5 times
larger than the minimum mass model. We also assume that during runaway migration planets up to Saturn’s mass undergo migration at the appropriate type I rate, which is implied approximately by the results of Masset & Papaloizou (2003).
We plot the growth times for three different evolutionary
models in Fig. 15. The dashed line represents the model with
standard opacity, the dashed-dotted line the model with one
tenth the standard opacity, and the dashed-dot-dot-dotted line
the model with opacity reduced by a factor of 100. During the
earliest phases of evolution the growth times of these models
are 3 × 108 , 3 × 107 and 3 × 106 years respectively. The two
models with largest opacity are thus unable to form giant planets within the disk life-time. Such systems will result in rock
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J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
Fig. 10. As in Fig. 6 but for models with a 5 M⊕ solid core which have opacities reduced by a factor of ten.
and ice cores forming that are unable to accrete significant gas
envelopes.
The lowest opacity model has a sufficiently low growth
time that it will be able to form a giant planet before disk dispersal. However, Fig. 15 shows that during the early stages of
evolution, while the planet mass is below 15 M⊕ , the growth
time is significantly longer than the type I migration time scale,
implying that the protoplanet will migrate into the central star
before forming a gas giant. This is a problem for all reasonable core instability models of gas giant formation, since there
exists a bottle neck for gas accretion while the planet mass is
relatively small, but massive enough to undergo quite rapid migration. If the core instability model is correct, then we are inevitably led to the conclusion that some process must operate
to prevent type I migration in a standard quiescent disk for at
least some protoplanets in order that gas giant planets can form.
Although many issues remain outstanding, a number of
processes may operate to prevent type I migration. Being the
result of a linear disk response it depends on the temperature and density structure of the disk and special features such
as rapid spatial variation of opacity may slow or stop migration (e.g. Menou & Goodman 2004). Recent simulations by
Nelson & Papaloizou (2004) and Nelson (2004) show that low
mass planets migrating in magnetised, turbulent accretion disks
undergo stochastic migration rather than monotonic inward
migration. This leads to a distribution of migration rates for
embedded planets, with some undergoing rapid inward migration, and others perhaps migrating outward or not at all. A
well defined direction of migration is likely to occur when the
planet mass is large enough to dominate over turbulent fluctuations, with simulations indicating that this is likely to arise
for planet masses exceeding ∼30 M⊕ . The occurrence of global
disk structures such as eccentric m = 1 modes are also capable of disrupting both type I (Papaloizou 2002) and type II
(Nelson 2003) migration, and if established within a disk are
likely to be long lived entities. Finally, low mass planets on eccentric orbits may undergo type I torque reversal (Papaloizou
& Larwood 2000). For an isolated planet the eccentricity is
quickly damped, but a system of mutually interacting planetary cores may be able to maintain eccentric orbits and hence
reduce or even prevent type I migration.
In light of these (and perhaps other) mechanisms for overcoming type I migration, which may operate in tandem rather
than in isolation, we make the assumption that for masses below Mpl = m0 30 M⊕ , type I migration is essentially ineffective for at least some protoplanets below that mass range, such
that a population of giant planet can form. It seems likely that
for planet masses larger than this, where the disk-planet interaction starts to become non linear, the ability of the planet to
impose itself on the disk will lead to inward migration being
re-established. When it does so Fig. 15 indicates it will be at
near the maximum rate. The transition from type A to type B
J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
261
Fig. 11. As in Fig. 6 but for models with a 5 M⊕ solid core accreting from the disk at a rate that is ten times slower. As a consequence of that it
takes about ten times longer to attain one Jupiter mass.
models is near to where Mpl = m0 . Note that m0 is likely to
depend on location and parameters in the disk making it uncertain whether the protoplanet undergoes some rapid inward
migration.
Returning to our discussion of the low opacity planet model
in Fig. 15, we can see that once the planet mass reaches 20 M⊕
and moves to a type B track, the growth time of the planet decreases dramatically down to a value that is determined by the
rate at which the protostellar disk can supply mass to the planet.
It is at this stage that we suppose that the protoplanet undergoes
a transition from being an extended structure in contact with its
Roche lobe, and accreting slowly from the disk, to a more compact protoplanet with a free surface that accretes rapidly from
the protostellar disk via a circumplanetary accretion disk. The
type B models presented in Figs. 13 and 14 suggest that these
compact models can accrete rapidly from the disk, and we specify a growth time of 3 × 103 years for this stage of growth in
Fig. 15, corresponding to the more rapid growth rates presented
in Fig. 7. Such a rate ensures that the planet can grow to become
a Jovian mass gas giant on a time scale shorter than any likely
migration time.
7.1.2. 15 M ⊕ core models
In Fig. 16 we present a schematic diagram of growth and migration times for planet models with 15 M⊕ cores. This figure is
very similar to Fig. 15. We have plotted evolutionary sequences
for just two planet models in Fig. 16, one with standard opacity
(dashed line) and one with three percent of the standard opacity (dashed-dotted line). During the earliest stages of accretion,
these models have growth times of 3 × 106 and 3 × 105 years,
respectively, which are comfortably within or below the range
of expected disk life-times. In the latter case, the formation is
very rapid, and illustrates the crucial role played by the opacity. However, the estimated quiescent disk migration time for
an object with a mass of 15−20 M⊕ is below 105 years, such
that even the lower opacity model is unable to form within the
expected type I migration time. We are again required to assume that type I migration is inoperative for some planets with
masses below m0 30 M⊕ .
As shown in Fig. 4, the growth time for the standard opacity model presented in Fig. 16 remains larger than the corresponding migration time for planet masses up to 34 M⊕ , at
which stage rapid gas accretion can ensue. Because such planet
models spend longer time at these higher masses they may be
more susceptible to undergoing a period of rapid migration either prior to or during the early stages of rapid gas accretion
than are the models with lower core masses. This may be related to an indication that extrasolar planets in systems with
high metallicity tend to be found at shorter periods commented
on by Santos et al. (2003). However, because of the small
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J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
Fig. 12. As in Fig. 6 but for models with a 5 M⊕ solid core accreting from the disk at a rate that is ten times slower and which have opacities a
factor of three smaller.
numbers involved, the statistical significance of such a trend
is not yet established.
The result of a rapid inward migration is that the planet
will move into the inner regions of the disk where: (i) the local reservoir of disk material is reduced relative to larger radii;
(ii) the disk aspect ratio H/R decreases making gap formation and a transition to type II migration easier (Papaloizou &
Terquem 1999). The result is likely to be a tendency for larger
and more dusty cores to produce a distribution of planets with
a greater bias toward low mass, short period objects.
7.2. Final protoplanet and circumplanetary disk
luminosity
We have seen that for model sequences of type B, the protoplanet luminosities are in general smaller and at best comparable to those expected from the circumplanetary disk. In the
cases with the most rapid accretion rate the difference is most
marked indicating that the accreting matter in general fails to
supply energy to the protoplanet. This feature causes the radius
to attain and remain near 2 × 1010 cm in most cases. It also
means that it is appropriate to regard protoplanets as undergoing disk accretion and able to accept all supplied material at
reasonable accretion rates once they cease to be enveloped by
the disk as in the type A case. The evolutionary time scales of
type A models are sensitive to the dust opacity, being directly
proportional and smaller for lower opacities.
The reason for the behaviour of type B models where they
fail to expand even at high accretion rates can be related to
some simple properties of barotropic stellar models that would
apply in the completely degenerate limit. For these P is a specified function of ρ and is related to the internal energy per unit
mass, U, through P = ρ2 (dU/dρ). Although the protoplanet
models are not of this type, they are similar enough to make
the discussion relevant.
The total energy is given by Eq. (12). For polytropes of
index n and U = nP/ρ, it is well known that (Chandrasekhar
1939)
E=−
2
(3 − n)GMpl
(5 − n)rs
,
(16)
(n−1)/(n−3)
. From this
while the mass radius relation is rs ∝ Mpl
if we consider a small mass increment dMpl , we get dE =
−(GMpl /rs )dMpl . But this change represents the energy lost
through disk and boundary layer accretion, leaving no input
for the polytrope explaining why it can remain of low luminosity. In fact the result is valid for any barotropic model and
can be shown to follow from the fact that at equilibrium the
change of energy is zero to first order in perturbations (the system can be thought of as being perturbed from equilibrium at a
J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
263
Fig. 13. As in Fig. 6 but for models with a 5 M⊕ solid core accreting from the disc at a rate that is 18 times faster and which have opacities a
factor of ten smaller.
slightly larger mass once added material is brought to the surface). In this way we can understand why models of type B can
remain of low luminosity, when compared to the fiducial value
of (GMpl /rs )dMpl /dt, under rapid accretion.
The expected circumplanetary disk or disk/protoplanet
boundary layer luminosity for a Jovian mass with radius 2 ×
1010 cm, and final accretion times in the range 105−6 y, lies in
the range ∼10−(3−4) L and the characteristic temperatures are
expected to be in the range 1000−2000 K.
8. Summary and conclusion
We have presented evolutionary models of giant protoplanets forming in protoplanetary disk. First, we have considered
planet models (type A) consisting of solid rock-ice cores surrounded by gaseous envelopes whose surface coincides with
the planet Hill sphere, and which accrete quasi-statically from
the surrounding protostellar disk. We have considered models
with 5 and 15 M⊕ cores, and have varied the dust opacity in the
envelope. For models with a 5 M⊕ core and standard opacity,
the time required for the planet to undergo rapid gas accretion
is ∼3 × 108 yr. This is longer than reasonable disk life-times
ranging between 3−30 Myr. Reductions in the dust opacity by
factors of 10 and 100 lead to models that require ∼3 × 107 and
3×106 yr, respectively, before rapid gas accretion ensues. Rapid
gas accretion occurs once these planets reach ∼18 M⊕ . A 15 M⊕
core planet model with standard opacity takes ∼3 × 106 yr before rapid gas accretion ensues. Models with dust opacity reduced by factors of 10 and 100 require ∼3 × 105 and ∼105 yr
before rapid gas accretion occurs. This arises once the planet
mass exceeds ∼35 M⊕ .
We present a second class of planet models (type B) where
the planet has a free surface, and accretes gas from a circumplanetary disk that is fed by the surrounding protostellar disk
at a specified rate. We find that these models can accrete gas
at any reasonable rate that may be supplied by the protostellar
disk without expansion.
We suggest that the earliest stages of giant planet formation are described by models of type A. For all such models,
the standard type I migration time is shorter than the accretion
time prior to rapid gas accretion. We suggest that type I migration is inoperative for at least some protoplanets with masses
below those for which disk-planet interactions becomes nonlinear (i.e. Mpl ∼ 30 M⊕ ), beyond which planets are more likely
to undergo rapid inward migration. In such a scenario planets with low mass cores and low opacity envelopes will have
a greater tendency to remain at larger radii up to the point of
rapid gas accretion. Those with more massive cores will tend
to undergo more rapid inward migration prior to or during rapid
gas accretion.
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J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
Fig. 14. As in Fig. 13 but for a sequence of models with the opacity reduction applied only for T < 1600 K.
Fig. 15. This diagram provides a schematic representation of the formation and migration time scales of planet models as a function of
protoplanet mass for core masses of 5 M⊕ . The range of plausible
protostellar disk life-times is indicated by the upper shaded region
spanning the times 3 × 106 −3 × 107 years. The migration time as a
function of planet mass is indicated by the solid line. A shaded region indicating the “danger zone” for rapid type I or runaway migration is also indicated. The growth time of protoplanets as a function
of planet mass is given for standard opacity (dashed line), 10 percent
opacity (dashed-dotted line), and 1 percent opacity (dashed-dot-dotdotted line). See text for discussion of this figure.
At the point of rapid gas accretion, we suppose that planets
contract within their Hill sphere, and are described by type B
Fig. 16. This diagram is similar to Fig. 15, except that it applies to
planet models with 15 M⊕ cores. The dashed line shows the growth
time as a function of mass for models with standard opacity. The
dashed-dotted line shows the growth time for models with 3 percent
opacity. See text for discussion of this figure.
models. The planets may now accrete at any rate supplied by
the protostellar disk, and can undergo rapid growth on a time
scale shorter than the migration time. If planets with low mass
cores tend to exist at larger radii during this stage, they may
make a rapid transition to Jovian mass objects, forming gaps
and entering a phase of slower type II migration. If planets
with larger mass cores have a tendency to undergo more rapid
J. C. B. Papaloizou and R. P. Nelson: Protoplanet models
inward migration, they may exist at smaller radii during the
rapid gas accretion phase. The disk is thinner here – such that
it is easier to form gaps, and the local reservoir of gas is smaller.
Such objects are more likely to end up with sub-Jovian masses.
We note that this general picture is likely to be blurred by
variations in disk parameters and life-times. But we also note
that the current extrasolar planet data shows a mass-period correlation in line with the simple ideas presented here (Zucker &
Mazeh 2002). Furthermore there is a hint of a correlation between host star metallicity and period such that higher metallicity stars appear to host shorter period planets (e.g. Santos et al.
2003). Such a correlation, while not statistically significant in
the data at present, may turn out to be as more data is accumulated and is accordingly a topic for scientific consideration
(e.g. Sozzetti 2004). We comment that a correlation of this type
might be expected if planetary core mass and envelope opacity
scale with the metallicity of the protoplanetary environment.
Acknowledgements. The hydrodynamic simulations performed here
were carried out using the QMUL HPC facility funded by the SRIF
initiative.
References
Beckwith, S. V. W., Sargent, A. I., Chini, R. S., & Guesten, R. 1990,
AJ, 99, 924
Bell, K. R., & Lin, D. N. C. 1994, ApJ, 427, 987
Bodenheimer, P., & Pollack, J. B. 1986, Icarus, 67, 391
Boss, A. P. 1998, ApJ, 503, 923
Cameron, A. G. W. 1978, Moon Planets, 18, 5
Chabrier, G., Saumon, D., Hubbard, W. B., & Lunine, J. I. 1992, ApJ,
391, 817
Chandrasekhar 1939, An Introduction to the Study of Stellar Structure
(New York: Dover Publications, Inc.)
Cox, J. P., & Giuli, R. T. 1968, Principles of Stellar Structure: Physical
Principles (New York: Gordon & Breach)
D’Angelo, G., Henning, T., & Kley, W. 2003, ApJ, 548, 599
D’Angelo, G., Kley, W., & Henning, T. 2003, ApJ, 586, 540
Ikoma, M., Nakazawa, K., & Emori, H. 2000, ApJ, 537, 10 103
265
Ida, S., & Makino, J. 1993, Icarus, 106, 210
Kley, W. 1999, MNRAS, 303, 696
Lissauer, J. J. 1993, ARA&A, 31, 129
Lubow, S., Siebert, M., & Artymowicz, P. 1999, ApJ, 526, 1001
Lynden-Bell, D., & Pringle, J. E. 1974, MNRAS, 168, 60
Masset, F. S., & Papaloizou, J. C. B. 2003, ApJ, 588, 494
Menou, K., & Goodman, J. 2004, ApJ, 606, 520
Mizuno, H. 1980, Prog. Theor. Phys., 64, 544
Nelson, R. P., Papaloizou, J. C. B., Masset, F. S., & Kley, W. 2000,
MNRAS, 318, 18
Nelson, R. P. 2003, MNRAS, 345, 233
Nelson, R. P., & Papaloizou, J. C. B. 2004, MNRAS, 350, 849
Nelson, R. P. 2004, in preparation
Papaloizou, J. C. B., & Terquem, C. 1999, ApJ, 521, 823
Papaloizou, J. C. B., & Larwood, J. D. 2000, MNRAS, 315, 823
Papaloizou, J. C. B. 2002, A&A, 388, 615
Podolak, M., Hubbard, W. B., & Pollack, J. B. 1993, in Protostars and
planets III, ed. E. H. Levy, & J. Lunine (Tucson: Univ. Arizona
Press), 1109
Pollack, J. B., Hubickyj, O., Bodenheimer, P., et al. 1996, Icarus, 124,
62
Rafikov, R. R. 2004, AJ, 128, 1348
Safronov, V. S. 1969, in Evoliutsiia doplanetnogo oblaka, Moscow
Santos, N. C., Israelian, G., Mayor, M., Rebolo, R., & Udry, S. 2003,
A&A, 398, 363
Saumon, D., & Guillot, T. 2004, ApJ, 609, 1170
Schwarzschild, M. 1958, Structure and Evolution of the Stars
(Princeton: Princeton Univ. Press)
Shakura, N. I., & Sunyaev, R. A. 1973, AAP, 24, 337
Sicilia-Aguilar, A., Hartmann, L. W., Briceno, C., Muzerolle, J., &
Calvet, N. 2004, AJ, 128, 805
Sozzetti, A. 2004, MNRAS, 354, 1194
Stevenson, D. J. 1982, Planet. Space. Sc., 30, 755
Tanaka, H., Takeuchi, T., & Ward, W. R. 2002, ApJ, 565, 1257
Thommes, E. W., Duncan, M. J., & Levison, H. F. 2003, Icarus, 161,
431
Ward, W. R. 1997, Icarus, 126, 261
Wetherill, G. W., & Stewart, G. R. 1989, Icarus, 77, 330
Ziegler, U., Yorke, H. W., & Kaisig, M. 1996, A&A, 305, 114
Zucker, S., & Mazeh, T. 2002, ApJ, 568, 113